modules over C[x]

Question

$M$ be a module over $\mathbb C[x]$ and let $T=$ {$m\in M: x^n\cdot m=0\ \text{for some}\ n>0$}. Is it true that $$dim_{\ \mathbb C(x)} M\otimes_{\mathbb C[x]}\mathbb C(x)=dim_{\ \mathbb C} (M/T)\otimes \mathbb C\ \text{?}$$ where $\mathbb C$ on the right is a $\mathbb C[x]$-module via $x=0.$ I can prove $\geq$ part and, for finitely generated $M$, the opposite inequality -- but not in general. Is it an equality for every countably generated $M$?

Answer 1

Take $M=\mathbb{C}[x]_x$. Then on the left you have

$$\text{dim}_{\mathbb{C}(x)}\mathbb{C}[x]_x\otimes \mathbb{C}(x)=1,$$

while on the right you have $$\text{dim}_{\mathbb{C}}\mathbb{C}[x]_x\otimes \mathbb{C}=0.$$

If you know about sheaves and affine varieties, you are likely to enjoy the algebro-geometric interpretation: the left hand side is the rank of the sheaf at the generic point, while on the right side we quotient out the torsion at 0 and take the rank. In the above example we get a strict inequality since the sheaf is supported away from 0.

Edit: Upper Semi-Continuity of the Rank of the Fibre of a Sheaf is, I think, the generic case of your argument for the opposite inequality in the finitely generated case.